Analyzing the dynamic behavior of the Gierer–Meinhardt model using finite difference method

In this paper, a numerical computation method for the Gierer–Meinhardt model in two-dimensional space diffusion with homogeneous Neumann boundary conditions, considering the interaction between activator and inhibitor substances, is proposed. First, a high-order compact finite difference scheme is constructed for the Gierer–Meinhardt model using the finite difference method. A fourth-order compact difference scheme is applied to the second-order spatial derivative terms, while the time derivative terms are discretized using Taylor series expansion and residual correction functions. Consequently, the difference scheme achieves fourth-order accuracy in space and second-order accuracy in time for the Gierer–Meinhardt model. In addition, the stability of the difference scheme is demonstrated using Fourier analysis. Finally, numerical simulations are conducted on the Gierer–Meinhardt model near its equilibrium point to explore the impact of the inhibitor degradation rate, denoted as E, on the pattern formation. The model exhibits distinct pattern structures with varying E, thereby revealing the relationship between tissue variability and pattern formation in biological systems.